Playbook
Complex Numbers
46 q - 1.00/paper - 33% HARD. Splits cleanly in two. Modulus, Argument and Polar Form (22 q) runs 18% HARD and is the half worth owning; Algebraic Equations, Locus and Cube Roots (24 q) is 46%.
- Questions in the bank
- 46
- q/paper (2024–25 shifts)
- 1.00
- Tagged HARD
- 33%
- Subtopics
- 2
Strand: Long Tail
When you’ll see it
An i in the expression — a modulus or argument asked for, a cube root of unity, or a locus described by a modulus condition.
How this chapter is tested
46 q, 1.00/paper, 33% HARD, and it splits about as cleanly as any chapter in the bank. Modulus, Argument, and Polar Form is 22 q at 18% HARD — the softest subtopic anywhere in the long tail. Algebraic Equations, Locus, and Cube Roots is 24 q at 46%.
That asymmetry IS the strategy. Own the polar half and treat the other as opportunistic: 18% HARD at one question a paper is about as close to free marks as the tail offers, and it is reachable with the identities of Trigonometry - I plus De Moivre's theorem.
The harder half runs largely on omega. Three facts about the cube roots of unity answer most of it: omega cubed is 1, 1 + omega + omega squared is 0, and powers of omega cycle with period 3 so any exponent can be reduced modulo 3. That is recall, not technique, which makes even the 46% corner tractable.
Locus questions are circles and lines in disguise — a condition of the form |z - a| = r is a circle of radius r centred at a. That is also where the cross-chapter extremum lives: the greatest and least modulus of z on such a disc is |a| plus or minus r, exactly the Circle chapter's distance-to-centre move, with no calculus and no differentiation of a modulus.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Algebra and the conjugate
Add, multiply, and rationalise a denominator by multiplying by the conjugate. z times its conjugate equals |z| squared — the identity that removes almost every fraction in the chapter.
Modulus and argument
|z| is the square root of (real part squared plus imaginary part squared). The argument comes from the ratio of the parts AND the quadrant of the point, never from the ratio alone.
Polar form and De Moivre
Write z as r(cos theta + i sin theta), then z^n is r^n (cos n theta + i sin n theta). This is what makes high powers and nth roots routine instead of expansive.
Cube roots of unity
omega^3 = 1 and 1 + omega + omega^2 = 0. Reduce every exponent modulo 3 first, then use the sum identity to collapse what remains.
Locus from a modulus or argument condition
|z - a| = r is a circle; |z - a| = |z - b| is the perpendicular bisector of the segment joining a and b; a fixed argument is a ray. Translate the condition into geometry before doing any algebra.
Extremum on a disc
Greatest and least |z| subject to |z - a| = r are |a| + r and |a| - r. Recognise it and the question is one subtraction, not an optimisation.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Argument from the ratio alone
The arctangent of the ratio gives a reference angle. Points in the second and third quadrants need pi added or subtracted, and the un-adjusted angle is always an option.
Modulus distributed over a sum
|z1 z2| = |z1| |z2| is true; |z1 + z2| = |z1| + |z2| is not, except in a degenerate case. The additive version is a planted distractor.
Principal argument out of range
The principal argument lies in (-pi, pi]. A value outside that interval must be shifted by 2 pi, and the unshifted value is offered.
Unreduced powers of omega
omega^4 is omega and omega^5 is omega squared. Leaving a high power unreduced produces an expression that looks unlike any option, which usually means the reduction was skipped.
Drill every complex numbers question
46 questions from the bank, scoped to 2 bundled subtopics.
Drill one subtopic at a time
The 2 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.